We are about to enter the twilight zone, that strange black hole in political time and space that appears no more than once every four years. It is known as the period of transition, and it starts a week from today, the time when the United States has not one president but two. One will be the president-elect, the other George Bush, in power for 12 more weeks in which he can do pretty much whatever he likes. Not only will he never again have to face voters, he won't even have to worry about damaging the prospects of his own party and its standard bearer (as if he has not damaged those enough already). From November 5 to January 20, he will exercise the freest, most unaccountable form of power the democratic world has to offer.

How Bush might use it is a question that gained new force at the weekend, when US forces crossed the Iraqi border into Syria to kill Abu Ghadiya, a man they said had been funnelling "foreign fighters" allied to al-Qaida into Iraq. That American move has touched off a round of intense head-scratching around the world, as foreign ministers and analysts ask each other the time-honoured diplomatic query: what did they mean by that? To which they add the post-Nov 4 question: and what does it tell us about how Bush plans to use his final days in the White House?

You can choose from two versions. Call the first the "no big deal" theory. It holds that the Sunday raid was no more than standard operational procedure in the war on terror. Sure, it meant violating the sovereignty of an independent nation state, but that's not so new: there was a similar incursion into Pakistan in September. Indeed, there may be more relevant precedents. A former official in the Bush administration confirmed to me yesterday that the US has lunged into Syrian territory several times before: it's just that Damascus chose to keep quiet. In which case, the interesting question is why the Syrians went public this time.

In this "no big deal" version, Abu Ghadiya was simply too irresistible a high-value target to let slip away. "They saw something they wanted to hit and they hit it," says one European diplomat resignedly. The most extreme version of this shoulder-shrugging account holds that the decision may not even have been taken at the political level, but in the field, by General David Petraeus. Not so implausible, since Bush in effect ceded command of the Iraq war to Petraeus a long while ago.

Nonsense, says the other school of thought. It is a massive deal to strike at a sovereign state in this way: in an earlier era, before 2001, we would have called it an act of war. Pakistan is no precedent, because in that case there was a degree of cooperation. Not now.

This was a deliberate act, calculated to send a series of messages. First, to the Syrians, reminding them who's boss in the region and strong-arming them to do more to crack down on al-Qaida.

Second, to the Europeans who have been moving towards a rapprochement with Damascus. Nicolas Sarkozy may have invited President Assad to Paris and David Miliband may have been hosting the Syrian foreign minister, Walid al-Muallem, in London this very Monday, 24 hours after the raid - but no matter. Bush gets to remind both these uppity Europeans who's in charge.

Third, the president could have been sending a message to his own administration. Perhaps this was a memo to his secretary of state, Condoleezza Rice, who had dared meet Muallem at the UN just last month in a meeting that apparently she requested. If so, it would fit with the pattern of wildly mixed signals that has emanated from the administration in recent months. Two days before Rice sat down with Muallem, for instance, Bush had used his UN address to denounce Syria as a state sponsor of terror. Might Sunday's raid have been the president's attempt to reassert himself against a senior staff all but denuded of its hawks? Rumsfeld, Bolton and Wolfowitz are long gone; the more emollient Robert Gates is at defence, widely tipped to continue under a President Obama. In these last days, Dick Cheney has only himself for company.

However we are meant to read it, the attack on Syria looks a lot like a parting shot from Bush, an end-of-the-movie reminder of what this long and bloody saga has been about. A small operation, causing eight deaths, it nevertheless captures much of the Bush ethos that has ruled the globe these past eight years. It was unilateral; it trampled on state sovereignty; and it relied on force as a first, not last, resort. As a souvenir of the Bush era, it would be hard to top.

But it may not be the final act. For we have not yet entered the twilight zone proper. That will come only when polls close next Tuesday. When the transition begins, all kinds of surprises are possible.

Spool back 20 years, to the dying days of the Reagan administration. In January 1989, the president officially recognised the PLO as the representatives of the Palestinian people. It was a farewell gift to Reagan's successor, George HW Bush: the old man took the flak so that the new president would not have to.

In December 1992, Bush himself proved rather less helpful to his replacement, saddling Bill Clinton with the deployment of US forces in Somalia, an episode whose humiliating conclusion badly hobbled Clinton thereafter.

Eight years ago, it was Clinton's turn. He sweated until his final hours in office trying to close a deal between Israel and the Palestinians, who seemed then to be just inches apart. The legacy was the Clinton parameters, still regarded as marking the basic contours of any future agreement for Israel-Palestine.

So what will emerge from the twilight of George W Bush? Most diplomats are bracing themselves. "They're not going to sleep," says one senior British official. The optimists hope for a repeat of Reagan and Clinton, something that helps Middle East peace. It's true that Rice and Bush have been eager for a breakthrough, if only to have a presidential legacy untainted by Iraq. Perhaps Israel and the Palestinians might initial a provisional document, proof that their labours since Bush's Annapolis summit of 2007 have not been entirely fruitless.

But the bad timing that has cursed the Middle East so often has struck once again. Israel is entering an interregnum of its own, following Tzipi Livni's failure to form a coalition. It's hard to believe an interim, caretaker administration could forge a peace deal.

That leaves other options. Bush could ape Reagan and decide to speak to Hamas. More likely would be a shift in policy that helps future peacemaking efforts: he might, for instance, declare that any changes to the 1967 borders must be equal, with Palestinians compensated inch for inch for any West Bank land conceded to Israel. Or he could look further afield in the region, contradicting himself and Sunday's raid, by reaching out to Syria. Or, as some hawks fear, he could step up the tentative dialogue with Iran. A symbolic gesture would be to open a US visa section in Tehran.

Of course, Bush may be thinking of a parting gift more in keeping with the record of the last eight years. He and Cheney might decide, what the hell, we have one last chance to whack Iran - and let the new guy clear up the mess. Not likely, but possible. For in the twilight zone, anything can happen.freedland@guardian.co.uk